# Pythagoras

*Mathematician and Philosopher*

*c. 582 b.c.e. – c. 500 b.c.e.*

Considered a mathematician, but foremost a philosopher, Pythagoras was a very important figure in mathematics, astronomy, musical theory, and in the world's history. However, little in the way of reliable record is known about his life and accomplishments. The accounts of Pythagoras inventing the musical scale, performing miracles, and announcing prophecies are probably only legend, and appear to have little historical foundation. Scholars generally agree only upon the main events in his life, and usually combine together discoveries by Pythagoras with those by his band of loyal followers.

Pythagoras established in what is now the southeastern coast of Italy a philosophical, political, and religious society whose members believed that the world could be explained using mathematics as based upon whole numbers and their ratios. Their motto was "All is number." Even the words philosophy (or "love of wisdom") and mathematics (or "that which is learned") is believed to have been first used (and defined) by the Pythagoreans.

Many Pythagorean beliefs (such as secrecy, vegetarianism, periods of food abstinence and silence, refusal to eat beans, refusal to wear animal skins, celibacy, self-examination, immortality, and reincarnation) were directed as "rules of life." The main focus of Pythagorean thought was ethics, developed primarily within philosophy, mathematics, music, and gymnastics. The beliefs of the society were that reality is mathematical; philosophy is used for spiritual purification; the soul is divine; and certain symbols possess mystical significance. Both men and women were permitted to become members. In fact, several female Pythagoreans became noted philosophers.*

***Aesara of Lucania was a Pythagorean philosopher known for her theory of the tripart soul, which she believed consisted of the mind, spiritedness, and desire.**

## How Pythagoreans Conceptualized Numbers

Pythagoreans believed that all relationships could be reduced to numbers in order to account for geometrical properties. This generalization originated from the observation that whenever the ratios of lengths of strings were whole numbers, harmonious tones were produced when these strings were vibrated.

The society studied properties of numbers that are familiar to modern mathematicians, such as even and odd numbers, prime and square numbers. From this viewpoint, the Pythagoreans developed the concept of number, which became their dominant principle of all proportion, order, and harmony in the universe.

The society also believed in such numerical properties as masculine or feminine, perfect or incomplete, and beautiful or ugly. These opposites, they believed, were found everywhere in nature, and the combination of them brought about the harmony of the world.

The primary belief of Pythagoreans in the sole existence of whole numbers was later challenged by their own findings, which proved the existence of "incommensurables," known today as **irrational numbers**. What is commonly called the "first crisis in mathematics" caused a scandal within the society, so serious that some members tried to suppress the knowledge of the existence of incommensurables.

The Pythagorean philosophy was dominated by the ideal that numbers were not only symbols of reality, but also were the final substance of real things, known as "number mysticism." They held, for example, that one is the point, two the line, three the surface, and four the solid. Seven was considered the destiny that dominates human life because infancy ends there, and also because the number was associated with the seven wandering stars. Moreover, Pythagoreans believed that maturity began at age 14, marriage occurred in the twenty-first year, and 70 years was the normal life span. Ten was identified as the "perfect number" because it was the sum of one, two, three, and

four.

## Pythagorean Contributions to Mathematics

The formalization of mathematics with the use of **axiomatic systems** was the most profound contribution that the Pythagorean society made to mathematics. Pythagoreans developed this significant concept by showing that arbitrary laws of empirical geometry could be proved as logical conclusions from a small number of axioms, or postulates. Typical of the developed axioms was "A straight line is the shortest distance between two points."

From these axioms, a number of **theorems** about the properties of points, lines, angles, curves, and planes could be logically deduced. These theorems include the famous Pythagorean theorem, which states that "the square of the **hypotenuse** of a right-angled triangle is equal to the sum of the squares of the other two sides." Another theorem states that the sum of the interior angles of any triangle is equal to the sum of two right angles.

## The Pythagorean Theorem

The Pythagoreans knew that any triangle whose sides were in the ratio 3:4:5 was a right-angled triangle. Their desire to find the mathematical harmonies of all things led them to prove the geometric theorem, today named for Pythagoras. The earlier Egyptians stated this theorem as an empirical relationship and, as far as is known today, the Pythagoreans were the first to prove it.

The Pythagorean (hypotenuse) theorem states that the square of the hypotenuse of a right-angle triangle (*c* ) is equal to the sum of the squares of the other two sides (*a* and *b* ), shown as *c* 2 = *a* 2 + *b* 2. The numbers 3, 4, and 5 are called Pythagorean numbers (5 2 = 3 2 + 4 2. or 25 = 9 + 16). However, the Pythagoreans did not consider the square on the hypotenuse to be that number (*c* ) multiplied by itself (*c* 2 ). Instead, it was conceptualized as a geometrical square (*C* ) constructed on the side of the hypotenuse, and that the sum of the areas of the two squares (*A* and *B* ) is equal to the area of the third square (*C* ), as shown below.

## Astronomy and the Pythagoreans

In astronomy, the Pythagoreans produced important advances in ancient scientific thought. They were the first to consider the Earth as a sphere revolving with the other planets and the Sun around a universal "central fire." Ten planets were believed to exist in order to produce the "magical" number of 10. This arrangement was explained as the harmonious arrangement of bodies in a complete sphere of reality based on a numerical pattern, calling it a "harmony of sphere." The Pythagoreans also recognized that the orbit of the Moon was **inclined** to the equator of the Earth, and were one of the first to accept that Venus was both the evening star and the morning star.

Even though much of the Pythagorean doctrine consisted of numerology and number mysticism, their influence in developing the idea that nature could be understood through mathematics and science was extremely important for studying and understanding the world in which we live.

see also Numbers: Abundant, Deficient, Perfect, and Amicable; Numbers, Forbidden and Superstitious; Numbers, Irrational; Numbers, Rational; Numbers, Whole; Triangle.

*William Arthur Atkins with*

*Philip Edward Koth*

## Bibliography

Boyer, Carl B. *A History of Mathematics,* 2 nd ed. New York: John Wiley & Sons, 1991.

O'Meara, Dominic J. *Pythagoras Revived: Mathematics and Philosophy in Late Antiquity.* New York: Clarendon Press, 1990.

Philip, James A. *Pythagoras and Early Pythagoreanism.* Toronto: University of Toronto Press, 1966.

## MAGIC OVER MATHEMATICS

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